What is the arclength of $f(t) = (tlnt,t/lnt)$ on $t in [1,e]$ ?

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Answer 1 · 2017-02-10T00:38:46

The arc length is infinite.

Notice that $lim_(trarr1^+)t/lnt=oo$ , since $ln(1)=0$ . This means that the arc length is infinite, as the function is asymptotic at $t=1$ .

We can also note that if we attempt to apply the arc length formula $L=int_a^bsqrt((dx/dt)^2+(dy/dt)^2)dt$ , the integral diverges.

Attached is a plot of the parametric graph for $tin(1,e]$ :

Desmos

What is the arclength of f(t) = (tlnt,t/lnt)f(t) = (tlnt,t/lnt) on t in [1,e]t in [1,e]?